Regression With Stata: Lesson 1 - Simple and Multiple Regression

8/9/2019 Regression With Stata: Lesson 1 - Simple and Multiple Regression

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Regression with Stata

Chapter 1 - Simple and Multiple Regression

Chapter Outline 1.0 Introduction

1.1 A First Regression Analysis 1.2 Examining Data

1.3 Simple linear regression

1.4 Multiple regression 1.5 Transforming variables

1.6 Summary

1.7 Self assessment 1.8 For more information

1.0 Introduction

This book is composed of four chapters covering a variety of topics about using Stata for regression. We should emphasize that this book is about "data

analysis" and that it demonstrates how Stata can be used for regression analysis, as opposed to a book that covers the statistical basis of multiple

regression. We assume that you have had at least one statistics course covering regression analysis and that you have a regression book that you can use

as a reference (see the Regression With Statapage and our Statistics Books for Loan pagefor recommended regression analysis books). This book isdesigned to apply your knowledge of regression, combine it with instruction on Stata, to perform, understand and interpret regression analyses.

This first chapter will cover topics in simple and multiple regression, as well as the supporting tasks that are important in preparing to analyze your data, e.g.,data checking, getting familiar with your data file, and examining the distribution of your variables. We will illustrate the basics of simple and multiple

regression and demonstrate the importance of inspecting, checking and verifying your data before accepting the results of your analysis. In general, we hopeto show that the results of your regression analysis can be misleading without further probing of your data, which could reveal relationships that a casual

analysis could overlook.

In this chapter, and in subsequent chapters, we will be using a data file that was created by randomly sampling 400 elementary schools from the CaliforniaDepartment of Education's API 2000 dataset. This data file contains a measure of school academic performance as well as other attributes of the

elementary schools, such as, class size, enrollment, poverty, etc.

You can access this data file over the web from within Stata with the Stata usecommand as shown below. Note:Do not type the leading dot in thecommand -- the dot is a convention to indicate that the statement is a Stata command.

use http://www.ats.ucla.edu/stat/stata/webbooks/reg/elemapi

Once you have read the file, you probably want to store a copy of it on your computer (so you don't need to read it over the web every time). Let's say youare using Windows and want to store the file in a folder called c:\regstata(you can choose a different name if you like). First, you can make this folder within

Stata using the mkdircommand.

mkdir c:\regstata

We can then change to that directory using the cdcommand.

cd c:\regstata

And then if you save the file it will be saved in the c:\regstatafolder. Let's save the file as elemapi.

save elemapi

Now the data file is saved as c:\regstata\elemapi.dtaand you could quit Stata and the data file would still be there. When you wish to use the file in the

future, you would just use the cdcommand to change to the c:\regstatadirectory (or whatever you called it) and then usethe elemapifile.

cd c:\regstatause elemapi

1.1 A First Regression Analysis

Let's dive right in and perform a regression analysis using the variables api00, acs_k3, mealsand full. These measure the academic performance of theschool (api00), the average class size in kindergarten through 3rd grade (acs_k3), the percentage of students receiving free meals (meals) - which is an

indicator of poverty, and the percentage of teachers who have full teaching credentials (full). We expect that better academic performance would be

associated with lower class size, fewer students receiving free meals, and a higher percentage of teachers having full teaching credentials. Below, we show

> statareg1.htm>stat >stata >webbooks >reg chapter1


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the Stata command for testing this regression model followed by the Stata output.

regress api00 acs_k3 meals full

Source | SS df MS Number of obs = 313-------------+------------------------------ F( 3, 309) = 213.41 Model | 2634884.26 3 878294.754 Prob > F = 0.0000 Residual | 1271713.21 309 4115.57673 R-squared = 0.6745-------------+------------------------------ Adj R-squared = 0.6713 Total | 3906597.47 312 12521.1457 Root MSE = 64.153

------------------------------------------------------------------------------ api00 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- acs_k3 | -2.681508 1.393991 -1.92 0.055 -5.424424 .0614073 meals | -3.702419 .1540256 -24.04 0.000 -4.005491 -3.399348 full | .1086104 .090719 1.20 0.232 -.0698947 .2871154 _cons | 906.7392 28.26505 32.08 0.000 851.1228 962.3555------------------------------------------------------------------------------

Let's focus on the three predictors, whether they are statistically significant and, if so, the direction of the relationship. The average class size (acs_k3,

b=-2.68), is not statistically significant at the 0.05 level (p=0.055), but only just so. The coefficient is negative which would indicate that larger class size isrelated to lower academic performance -- which is what we would expect. Next, the effect of meals(b=-3.70, p=.000) is significant and its coefficient is

negative indicating that the greater the proportion students receiving free meals, the lower the academic performance. Please note, that we are not saying

that free meals are causing lower academic performance. The mealsvariable is highly related to income level and functions more as a proxy for poverty.Thus, higher levels of poverty are associated with lower academic performance. This result also makes sense. Finally, the percentage of teachers with full

credentials (full, b=0.11, p=.232) seems to be unrelated to academic performance. This would seem to indicate that the percentage of teachers with fullcredentials is not an important factor in predicting academic performance -- this result was somewhat unexpected.

Should we take these results and write them up for publication? From these results, we would conclude that lower class sizes are related to higherperformance, that fewer students receiving free meals is associated with higher performance, and that the percentage of teachers with full credentials wasnot related to academic performance in the schools. Before we write this up for publication, we should do a number of checks to make sure we can firmly

stand behind these results. We start by getting more familiar with the data file, doing preliminary data checking, looking for errors in the data.

1.2 Examining data

First, let's use the describecommand to learn more about this data file. We can verify how many observations it has and see the names of the variables it

contains. To do this, we simply type

describe

Contains data from http://www.ats.ucla.edu/stat/stata/webbooks/reg/elemapi.dta obs: 400vars: 21 25 Feb 2001 16:58size: 14,800 (92.3% of memory free)

------------------------------------------------------------------------------- storage display valuevariable name type format label variable label-------------------------------------------------------------------------------snum int %9.0g school numberdnum int %7.0g dname district numberapi00 int %6.0g api 2000api99 int %6.0g api 1999growth int %6.0g growth 1999 to 2000meals byte %4.0f pct free mealsell byte %4.0f english language learnersyr_rnd byte %4.0f yr_rnd year round schoolmobility byte %4.0f pct 1st year in schoolacs_k3 byte %4.0f avg class size k-3acs_46 byte %4.0f avg class size 4-6not_hsg byte %4.0f parent not hsghsg byte %4.0f parent hsgsome_col byte %4.0f parent some college

col_grad byte %4.0f parent college gradgrad_sch byte %4.0f parent grad schoolavg_ed float %9.0g avg parent edfull float %4.0f pct full credentialemer byte %4.0f pct emer credentialenroll int %9.0g number of studentsmealcat byte %18.0g mealcat Percentage free meals in 3 categories-------------------------------------------------------------------------------Sorted by: dnum

We will not go into all of the details of this output. Note that there are 400 observations and 21 variables. We have variables about academic performance in

2000 and 1999 and the change in performance, api00, api99and growthrespectively. We also have various characteristics of the schools, e.g., class size,

parents education, percent of teachers with full and emergency credentials, and number of students. Note that when we did our original regression analysisit said that there were 313 observations, but the describecommand indicates that we have 400 observations in the data file.


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If you want to learn more about the data file, you could listall or some of the observations. For example, below we listthe first five observations.

list in 1/5

Observation 1

snum 906 dnum 41 api00 693 api99 600 growth 93 meals 67 ell 9 yr_rnd No mobility 11 acs_k3 16 acs_46 22 not_hsg 0 hsg 0 some_col 0 col_grad 0

grad_sch 0 avg_ed . full 76.00 emer 24 enroll 247 mealcat 47-80% free

Observation 2

snum 889 dnum 41 api00 570 api99 501 growth 69 meals 92 ell 21 yr_rnd No mobility 33 acs_k3 15 acs_46 32 not_hsg 0 hsg 0 some_col 0 col_grad 0 grad_sch 0 avg_ed . full 79.00 emer 19 enroll 463 mealcat 81-100% free

Observation 3

snum 887 dnum 41 api00 546

api99 472 growth 74 meals 97 ell 29 yr_rnd No mobility 36 acs_k3 17 acs_46 25 not_hsg 0 hsg 0 some_col 0 col_grad 0 grad_sch 0 avg_ed . full 68.00 emer 29 enroll 395 mealcat 81-100% free

Observation 4

snum 876 dnum 41 api00 571 api99 487 growth 84 meals 90 ell 27 yr_rnd No mobility 27 acs_k3 20 acs_46 30 not_hsg 36 hsg 45 some_col 9 col_grad 9 grad_sch 0 avg_ed 1.91 full 87.00 emer 11 enroll 418 mealcat 81-100% free

Observation 5

snum 888 dnum 41 api00 478 api99 425 growth 53 meals 89 ell 30 yr_rnd No mobility 44 acs_k3 18 acs_46 31 not_hsg 50 hsg 50 some_col 0 col_grad 0 grad_sch 0 avg_ed 1.5 full 87.00 emer 13 enroll 520 mealcat 81-100% free

This takes up lots of space on the page, but does not give us a lot of information. Listing our data can be very helpful, but it is more helpful if you listjust thevariables you are interested in. Let's listthe first 10 observations for the variables that we looked at in our first regression analysis.

list api00 acs_k3 meals full in 1/10

api00 acs~3 meals full1. 693 16 67 76.002. 570 15 92 79.003. 546 17 97 68.004. 571 20 90 87.005. 478 18 89 87.006. 858 20 . 100.007. 918 19 . 100.008. 831 20 . 96.009. 860 20 . 100.00

10. 737 21 29 96.00

We see that among the first 10 observations, we have four missing values for meals. It is likely that the missing data for mealshad something to do with the

fact that the number of observations in our first regression analysis was 313 and not 400.


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Another useful tool for learning about your variables is the codebookcommand. Let's do codebookfor the variables we included in the regression analysis,

as well as the variable yr_rnd. We have interspersed some comments on this output in [square brackets and in bold].

codebook api00 acs_k3 meals full yr_rnd

api00 ---------------------------------------------------------------- api 2000 type: numeric (int)

range: [369,940] units: 1 unique values: 271 coded missing: 0 / 400

mean: 647.622 std. dev: 142.249

percentiles: 10% 25% 50% 75% 90% 465.5 523.5 643 762.5 850[the api scores don't have any missing values, and range from 369-940][this makes sense since the api scores can range from 200 to 1000]

acs_k3 ----------------------------------------------------- avg class size k-3 type: numeric (byte)

range: [-21,25] units: 1 unique values: 14 coded missing: 2 / 400

mean: 18.5477 std. dev: 5.00493

percentiles: 10% 25% 50% 75% 90% 17 18 19 20 21[the average class size ranges from -21 to 25 and 2 are missing.][it seems odd for a class size to be -21]

meals ---------------------------------------------------------- pct free meals type: numeric (byte)


mean: 71.9937 std. dev: 24.3856

percentiles: 10% 25% 50% 75% 90% 33 57 77 93 99[the percent receiving free meals ranges from 6 to 100, but 85 are missing][this seems like a large number of missing values!]

full ------------------------------------------------------ pct full credential type: numeric (float)

range: [.42,100] units: .01 unique values: 92 coded missing: 0 / 400

mean: 66.0568 std. dev: 40.2979

percentiles: 10% 25% 50% 75% 90%67 .95 87 97 100[The percent credentialed ranges from .42 to 100 with no missing]

yr_rnd ------------------------------------------------------ year round school

type: numeric (byte) label: yr_rnd


tabulation: Freq. Numeric Label 308 0 No 92 1 Yes[the variable yr_rnd is coded 0=No (not year round) and 1=Yes (year round)][308 are non-year round and 92 are year round, and none are missing]

The codebook command has uncovered a number of peculiarities worthy of further examination. Let's use the summarizecommand to learn more aboutthese variables. As shown below, the summarizecommand also reveals the large number of missing values for meals(400 - 315 = 85) and we see the

unusual minimum for acs_k3of -21.


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summarize api00 acs_k3 meals full

Variable | Obs Mean Std. Dev. Min Max-------------+----------------------------------------------------- api00 | 400 647.6225 142.249 369 940 acs_k3 | 398 18.54774 5.004933 -21 25 meals | 315 71.99365 24.38557 6 100 full | 400 66.0568 40.29793 .42 100

Let's get a more detailed summary for acs_k3. In Stata, the comma after the variable list indicates that options follow, in this case, the option is detail. Asyou can see below, the detailoption gives you the percentiles, the four largest and smallest values, measures of central tendency and variance, etc. Note

that summarize, and other commands, can be abbreviated: we could have typed sum acs_k3, d.

summarize acs_k3, detail

avg class size k-3------------------------------------------------------------- Percentiles Smallest1% -20 -215% 16 -2110% 17 -21 Obs 39825% 18 -20 Sum of Wgt. 398

50% 19 Mean 18.54774 Largest Std. Dev. 5.00493375% 20 2390% 21 23 Variance 25.0493595% 21 23 Skewness -7.078785

99% 23 25 Kurtosis 55.33497

It seems as though some of the class sizes somehow became negative, as though a negative sign was incorrectly typed in front of them. Let's do atabulateof class size to see if this seems plausible.

tabulate acs_k3avg class |

size k-3 | Freq. Percent Cum.------------+----------------------------------- -21 | 3 0.75 0.75 -20 | 2 0.50 1.26 -19 | 1 0.25 1.51 14 | 2 0.50 2.01 15 | 1 0.25 2.26 16 | 14 3.52 5.78 17 | 20 5.03 10.80 18 | 64 16.08 26.88 19 | 143 35.93 62.81 20 | 97 24.37 87.19 21 | 40 10.05 97.24 22 | 7 1.76 98.99 23 | 3 0.75 99.75 25 | 1 0.25 100.00------------+----------------------------------- Total | 398 100.00

Indeed, it seems that some of the class sizes somehow got negative signs put in front of them. Let's look at the school and district number for theseobservations to see if they come from the same district. Indeed, they all come from district 140.

list snum dnum acs_k3 if acs_k3 < 0

snum dnum acs~337. 602 140 -2196. 600 140 -20173. 595 140 -21223. 596 140 -19229. 611 140 -20282. 592 140 -21

Let's look at all of the observations for district 140.

list dnum snum api00 acs_k3 meals full if dnum == 140

dnum snum api00 acs~3 meals full


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37. 140 602 864 -21 . 100.0096. 140 600 843 -20 . 91.00173. 140 595 713 -21 63 92.00223. 140 596 800 -19 . 94.00229. 140 611 857 -20 . 100.00282. 140 592 804 -21 . 97.00

All of the observations from district 140 seem to have this problem. When you find such a problem, you want to go back to the original source of the data toverify the values. We have to reveal that we fabricated this error for illustration purposes, and that the actual data had no such problem. Let's pretend that we

checked with district 140 and there was a problem with the data there, a hyphen was accidentally put in front of the class sizes making them negative. Wewill make a note to fix this! Let's continue checking our data.

Let's take a look at some graphical methods for inspecting data. For each variable, it is useful to inspect them using a histogram, boxplot, and stem-and-leafplot. These graphs can show you information about the shape of your variables better than simple numeric statistics can. We already know about theproblem with acs_k3, but let's see how these graphical methods would have revealed the problem with this variable.

First, we show a histogram for acs_k3. This shows us the observations where the average class size is negative.

histogram acs_k3

Likewise, a boxplot would have called these observations to our attention as well. You can see the outlying negative observations way at the bottom of theboxplot.

graph box acs_k3

Finally, a stem-and-leaf plot would also have helped to identify these observations. This plot shows the exact values of the observations, indicating thatthere were three -21s, two -20s, and one -19.

stem acs_k3

Stem-and-leaf plot for acs_k3 (avg class size k-3)

-2* | 11100-1. | 9


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-1s |-1f |-1t |-1* |-0. |-0s |-0f |-0t |-0* |0* |0t |0f |

0s |0. |1* |1t |1f | 445

1s | 6666666666666677777777777777777777 1. | 88888888888888888888888888888888888888888888888888888888888888 ... (207) 2* | 00000000000000000000000000000000000000000000000000000000000000 ... (137) 2t | 2222222333 2f | 5

We recommend plotting all of these graphs for the variables you will be analyzing. We will omit, due to space considerations, showing these graphs for all of

the variables. However, in examining the variables, the stem-and-leaf plot for fullseemed rather unusual. Up to now, we have not seen anything problematicwith this variable, but look at the stem and leaf plot for fullbelow. It shows 104 observations where the percent with a full credential is less than one. This is

over 25% of the schools, and seems very unusual.

stem fullStem-and-leaf plot for full (pct full credential)

full rounded to nearest multiple of .1plot in units of .1

0** | 04,04,05,05,05,05,05,05,05,05,05,05,06,06,06,06,06,06,06,06, ... (104) 0** |

0** |0** |0** |1** |1** |1** |1** |1** |2** |2** |2** |2** |2** |3** |3** |3** |3** | 70

3** |4** | 10

4** |4** | 40,40,50,50

4** | 60 4** | 80 5** |

5** | 30 5** |

5** | 70 5** | 80,80,80,90 6** | 10 6** | 30,30 6** | 40,50 6** |

6** | 80,80,90,90,90 7** | 00,10,10,10 7** | 20,30,30 7** | 40,50,50,50,50 7** | 60,60,60,60,70,70 7** | 80,80,80,80,90,90,90 8** | 00,00,00,00,00,00,00,00,00,00,10,10,10,10 8** | 20,20,20,30,30,30,30,30,30,30,30,30 8** | 40,40,40,40,50,50,50,50,50,50,50,50 8** | 60,60,60,60,60,70,70,70,70,70,70,70,70,70,70,70,70


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8** | 80,80,80,80,80,80,90,90,90,90,90 9** | 00,00,00,00,00,00,00,00,00,10,10,10,10,10,10,10,10 9** | 20,20,20,20,20,20,20,30,30,30,30,30,30,30,30,30,30,30,30 9** | 40,40,40,40,40,40,40,40,40,40,50,50,50,50,50,50,50,50,50,50, ... (27) 9** | 60,60,60,60,60,60,60,60,60,60,60,60,60,60,60,60,60,70,70,70, ... (28) 9** | 80,80,80,80,80,80,80,80,80 10** | 00,00,00,00,00,00,00,00,00,00,00,00,00,00,00,00,00,00,00,00, ... (81)

Let's look at the frequency distribution of fullto see if we can understand this better. The values go from 0.42 to 1.0, then jump to 37 and go up from there.It appears as though some of the percentages are actually entered as proportions, e.g., 0.42 was entered instead of 42 or 0.96 which really should have

been 96.

tabulate full

pct full |credential | Freq. Percent Cum.------------+----------------------------------- 0.42 | 1 0.25 0.25 0.45 | 1 0.25 0.50 0.46 | 1 0.25 0.75 0.47 | 1 0.25 1.00 0.48 | 1 0.25 1.25 0.50 | 3 0.75 2.00 0.51 | 1 0.25 2.25 0.52 | 1 0.25 2.50 0.53 | 1 0.25 2.75 0.54 | 1 0.25 3.00 0.56 | 2 0.50 3.50

0.57 | 2 0.50 4.00 0.58 | 1 0.25 4.25 0.59 | 3 0.75 5.00 0.60 | 1 0.25 5.25 0.61 | 4 1.00 6.25 0.62 | 2 0.50 6.75 0.63 | 1 0.25 7.00 0.64 | 3 0.75 7.75 0.65 | 3 0.75 8.50 0.66 | 2 0.50 9.00 0.67 | 6 1.50 10.50 0.68 | 2 0.50 11.00 0.69 | 3 0.75 11.75 0.70 | 1 0.25 12.00 0.71 | 1 0.25 12.25 0.72 | 2 0.50 12.75 0.73 | 6 1.50 14.25 0.75 | 4 1.00 15.25 0.76 | 2 0.50 15.75 0.77 | 2 0.50 16.25 0.79 | 3 0.75 17.00 0.80 | 5 1.25 18.25 0.81 | 8 2.00 20.25 0.82 | 2 0.50 20.75 0.83 | 2 0.50 21.25 0.84 | 2 0.50 21.75 0.85 | 3 0.75 22.50 0.86 | 2 0.50 23.00 0.90 | 3 0.75 23.75 0.92 | 1 0.25 24.00 0.93 | 1 0.25 24.25 0.94 | 2 0.50 24.75 0.95 | 2 0.50 25.25 0.96 | 1 0.25 25.50 1.00 | 2 0.50 26.00

37.00 | 1 0.25 26.25 41.00 | 1 0.25 26.50 44.00 | 2 0.50 27.00 45.00 | 2 0.50 27.50 46.00 | 1 0.25 27.75 48.00 | 1 0.25 28.00 53.00 | 1 0.25 28.25 57.00 | 1 0.25 28.50 58.00 | 3 0.75 29.25 59.00 | 1 0.25 29.50 61.00 | 1 0.25 29.75 63.00 | 2 0.50 30.25 64.00 | 1 0.25 30.50 65.00 | 1 0.25 30.75 68.00 | 2 0.50 31.25 69.00 | 3 0.75 32.00


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70.00 | 1 0.25 32.25 71.00 | 3 0.75 33.00 72.00 | 1 0.25 33.25 73.00 | 2 0.50 33.75 74.00 | 1 0.25 34.00 75.00 | 4 1.00 35.00 76.00 | 4 1.00 36.00 77.00 | 2 0.50 36.50 78.00 | 4 1.00 37.50 79.00 | 3 0.75 38.25 80.00 | 10 2.50 40.75 81.00 | 4 1.00 41.75

82.00 | 3 0.75 42.50 83.00 | 9 2.25 44.75 84.00 | 4 1.00 45.75 85.00 | 8 2.00 47.75 86.00 | 5 1.25 49.00 87.00 | 12 3.00 52.00 88.00 | 6 1.50 53.50 89.00 | 5 1.25 54.75 90.00 | 9 2.25 57.00 91.00 | 8 2.00 59.00 92.00 | 7 1.75 60.75 93.00 | 12 3.00 63.75 94.00 | 10 2.50 66.25 95.00 | 17 4.25 70.50 96.00 | 17 4.25 74.75 97.00 | 11 2.75 77.50 98.00 | 9 2.25 79.75 100.00 | 81 20.25 100.00

------------+----------------------------------- Total | 400 100.00

Let's see which district(s) these data came from.

tabulate dnum if full


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We have identified three problems in our data. There are numerous missing values for meals, there were negatives accidentally inserted before some of theclass sizes (acs_k3) and over a quarter of the values for fullwere proportions instead of percentages. The corrected version of the data is called elemapi2.

Let's use that data file and repeat our analysis and see if the results are the same as our original analysis. First, let's repeat our original regression analysis

below.

regress api00 acs_k3 meals full


------------------------------------------------------------------------------ api00 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- acs_k3 | -2.681508 1.393991 -1.92 0.055 -5.424424 .0614073 meals | -3.702419 .1540256 -24.04 0.000 -4.005491 -3.399348 full | .1086104 .090719 1.20 0.232 -.0698947 .2871154 _cons | 906.7392 28.26505 32.08 0.000 851.1228 962.3555------------------------------------------------------------------------------

Now, let's use the corrected data file and repeat the regression analysis. We see quite a difference in the results! In the original analysis (above), acs_k3was nearly significant, but in the corrected analysis (below) the results show this variable to be not significant, perhaps due to the cases where class size wasgiven a negative value. Likewise, the percentage of teachers with full credentials was not significant in the original analysis, but is significant in the corrected

analysis, perhaps due to the cases where the value was given as the proportion with full credentials instead of the percent. Also, note that the corrected

analysis is based on 398 observations instead of 313 observations, due to getting the complete data for the mealsvariable which had lots of missing values.

use http://www.ats.ucla.edu/stat/stata/webbooks/reg/elemapi2regress api00 acs_k3 meals full


------------------------------------------------------------------------------ api00 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- acs_k3 | -.7170622 2.238821 -0.32 0.749 -5.118592 3.684468 meals | -3.686265 .1117799 -32.98 0.000 -3.906024 -3.466505 full | 1.327138 .2388739 5.56 0.000 .857511 1.796765 _cons | 771.6581 48.86071 15.79 0.000 675.5978 867.7184------------------------------------------------------------------------------

From this point forward, we will use the corrected, elemapi2, data file. You might want to save this on your computer so you can use it in future analyses.

save elemapi2


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So far we have covered some topics in data checking/verification, but we have not really discussed regression analysis itself. Let's now talk more about

performing regression analysis in Stata.

1.3 Simple Linear Regression

Let's begin by showing some examples of simple linear regression using Stata. In this type of regression, we have only one predictor variable. This variable

may be continuous, meaning that it may assume all values within a range, for example, age or height, or it may be dichotomous, meaning that the variablemay assume only one of two values, for example, 0 or 1. The use of categorical variables with more than two levels will be covered in Chapter 3. There is

only one response or dependent variable, and it is continuous.

In Stata, the dependent variable is listed immediately after the regresscommand followed by one or more predictor variables. Let's examine the relationship

between the size of school and academic performance to see if the size of the school is related to academic performance. For this example, api00is the

dependent variable and enroll is the predictor.

regress api00 enroll


------------------------------------------------------------------------------ api00 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- enroll | -.1998674 .0298512 -6.70 0.000 -.2585532 -.1411817 _cons | 744.2514 15.93308 46.71 0.000 712.9279 775.5749------------------------------------------------------------------------------

Let's review this output a bit more carefully. First, we see that the F-test is statistically significant, which means that the model is statistically significant. The

R-squared of .1012 means that approximately 10% of the variance of api00is accounted for by the model, in this case, enroll. The t-test for enrollequals

-6.70, and is statistically significant, meaning that the regression coefficient for enroll is significantly different from zero. Note that (-6.70)2= 44.89, which is

the same as the F-statistic (with some rounding error). The coefficient for enrollis -.1998674, or approximately -.2, meaning that for a one unit increase inenroll, we would expect a .2-unit decrease in api00. In other words, a school with 1100 students would be expected to have an api score 20 units lower than

a school with 1000 students. The constant is 744.2514, and this is the predicted value when enrollequals zero. In most cases, the constant is not very

interesting. We have prepared an annotated outputwhich shows the output from this regression along with an explanation of each of the items in it.

In addition to getting the regression table, it can be useful to see a scatterplot of the predicted and outcome variables with the regression line plotted. After

you run a regression, you can create a variable that contains the predicted values using the predictcommand. You can get these values at any point afteryou run a regresscommand, but remember that once you run a new regression, the predicted values will be based on the most recent regression. To create

predicted values you just type predict and the name of a new variable Stata will give you the fitted values. For this example, our new variable name will be fv,

so we will type

predict fv

(option xb assumed; fitted values)

If we use the listcommand, we see that a fitted value has been generated for each observation.

list api00 fv in 1/10

api00 fv1. 369 542.58512. 386 671.49963. 386 661.70624. 387 541.78575. 387 592.15236. 394 618.53487. 397 543.5845

8. 406 604.54419. 411 645.516910. 412 491.619

Below we can show a scatterplot of the outcome variable, api00and the predictor, enroll.

scatter api00 enroll


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We can combine scatter with lfit to show a scatterplot with fitted values.

twoway (scatter api00 enroll) (lfit api00 enroll)

As you see, some of the points appear to be outliers. If you use the mlabel(snum) option on the scatter command, you can see the school number for

each point. This allows us to see, for example, that one of the outliers is school 2910.

twoway (scatter api00 enroll, mlabel(snum)) (lfit api00 enroll)

As we saw earlier, the predictcommand can be used to generate predicted (fitted) values after running regress. You can also obtain residuals by using the

predictcommand followed by a variable name, in this case e, with the residualoption.


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predict e, residual

This command can be shortened to predict e, residor even predict e, r. The table below shows some of the other values can that be created with thepredictoption.

Value to be created Option after Predict--------------------------------------------------- --------------------predicted values of y (y is the dependent variable) no option neededresiduals residstandardized residuals rstandardstudentized or jackknifed residuals rstudentleverage lev or hatstandard error of the residual stdrCook's D cooksdstandard error of predicted individual y stdfstandard error of predicted mean y stdp

1.4 Multiple Regression

Now, let's look at an example of multiple regression, in which we have one outcome (dependent) variable and multiple predictors. Before we begin with our

next example, we need to make a decision regarding the variables that we have created, because we will be creating similar variables with our multiple

regression, and we don't want to get the variables confused. For example, in the simple regression we created a variable fvfor our predicted (fitted) valuesand efor the residuals. If we want to create predicted values for our next example we could call the predicted value something else, e.g., fv_mr, but this

could start getting confusing. We could drop the variables we have created, using drop fv e. Instead, let's clear out the data in memory and usethe

elemapi2data file again. When we start new examples in future chapters, we will clear out the existing data file and use the file again to start fresh.

clearuse http://www.ats.ucla.edu/stat/stata/webbooks/reg/elemapi2

For this multiple regression example, we will regress the dependent variable, api00, on all of the predictor variables in the data set.

regress api00 ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll


------------------------------------------------------------------------------ api00 | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+----------------------------------------------------------------

ell | -.8600707 .2106317 -4.08 0.000 -1.274203 -.4459382 meals | -2.948216 .1703452 -17.31 0.000 -3.28314 -2.613293 yr_rnd | -19.88875 9.258442 -2.15 0.032 -38.09218 -1.68531 mobility | -1.301352 .4362053 -2.98 0.003 -2.158995 -.4437089 acs_k3 | 1.3187 2.252683 0.59 0.559 -3.1104 5.747801 acs_46 | 2.032456 .7983213 2.55 0.011 .462841 3.602071 full | .609715 .4758205 1.28 0.201 -.3258169 1.545247 emer | -.7066192 .6054086 -1.17 0.244 -1.89694 .4837018 enroll | -.012164 .0167921 -0.72 0.469 -.0451798 .0208517 _cons | 778.8305 61.68663 12.63 0.000 657.5457 900.1154------------------------------------------------------------------------------

Let's examine the output from this regression analysis. As with the simple regression, we look to the p-value of the F-test to see if the overall model issignificant. With a p-value of zero to four decimal places, the model is statistically significant. The R-squared is 0.8446, meaning that approximately 84% of

the variability of api00is accounted for by the variables in the model. In this case, the adjusted R-squared indicates that about 84% of the variability of api00

is accounted for by the model, even after taking into account the number of predictor variables in the model. The coefficients for each of the variablesindicates the amount of change one could expect in api00given a one-unit change in the value of that variable, given that all other variables in the model are

held constant. For example, consider the variable ell. We would expect a decrease of 0.86 in the api00score for every one unit increase in ell, assumingthat all other variables in the model are held constant. The interpretation of much of the output from the multiple regression is the same as it was for the

simple regression. We have prepared an annotated outputthat more thoroughly explains the output of this multiple regression analysis.

You may be wondering what a 0.86 change in ellreally means, and how you might compare the strength of that coefficient to the coefficient for anothervariable, say meals. To address this problem, we can add an option to the regresscommand called beta, which will give us the standardized regression

coefficients. The beta coefficients are used by some researchers to compare the relative strength of the various predictors within the model. Because the

beta coefficients are all measured in standard deviations, instead of the units of the variables, they can be compared to one another. In other words, the betacoefficients are the coefficients that you would obtain if the outcome and predictor variables were all transformed standard scores, also called z-scores,

before running the regression.

regress api00 ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll, beta

Source | SS df MS Number of obs = 395-------------+------------------------------ F( 9, 385) = 232.41 Model | 6740702.01 9 748966.89 Prob > F = 0.0000


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Residual | 1240707.78 385 3222.61761 R-squared = 0.8446-------------+------------------------------ Adj R-squared = 0.8409 Total | 7981409.79 394 20257.3852 Root MSE = 56.768

------------------------------------------------------------------------------ api00 | Coef. Std. Err. t P>|t| Beta-------------+---------------------------------------------------------------- ell | -.8600707 .2106317 -4.08 0.000 -.1495771 meals | -2.948216 .1703452 -17.31 0.000 -.6607003 yr_rnd | -19.88875 9.258442 -2.15 0.032 -.0591404 mobility | -1.301352 .4362053 -2.98 0.003 -.0686382 acs_k3 | 1.3187 2.252683 0.59 0.559 .0127287

acs_46 | 2.032456 .7983213 2.55 0.011 .0549752 full | .609715 .4758205 1.28 0.201 .0637969 emer | -.7066192 .6054086 -1.17 0.244 -.0580132 enroll | -.012164 .0167921 -0.72 0.469 -.0193554 _cons | 778.8305 61.68663 12.63 0.000 .------------------------------------------------------------------------------

Because the coefficients in the Beta column are all in the same standardized units you can compare these coefficients to assess the relative strength of each

of the predictors. In this example, mealshas the largest Beta coefficient, -0.66 (in absolute value), and acs_k3has the smallest Beta, 0.013. Thus, a one

standard deviation increase in mealsleads to a 0.66 standard deviation decrease in predicted api00, with the other variables held constant. And, a onestandard deviation increase in acs_k3, in turn, leads to a 0.013 standard deviation increase in predicted api00with the other variables in the model held

constant.

In interpreting this output, remember that the difference between the numbers listed in the Coef. column and the Beta column is in the units of measurement.For example, to describe the raw coefficient for ellyou would say "A one-unit decrease in ellwould yield a .86-unit increase in the predicted api00."

However, for the standardized coefficient (Beta) you would say, "A one standard deviation decrease in ellwould yield a .15 standard deviation increase in thepredicted api00."

The listcoefcommand gives more extensive output regarding standardized coefficients. It is not part of Stata, but you can download it over the internet likethis.

findit listcoef

and then follow the instructions (see also How can I use the findit command to search for programs and get additional help?for more information about using

findit). Now that we have downloaded listcoef, we can run it like this.

listcoef

regress (N=395): Unstandardized and Standardized Estimates

Observed SD: 142.32844SD of Error: 56.768104

--------------------------------------------------------------------------- api00 | b t P>|t| bStdX bStdY bStdXY SDofX---------+----------------------------------------------------------------- ell | -0.86007 -4.083 0.000 -21.2891 -0.0060 -0.1496 24.7527 meals | -2.94822 -17.307 0.000 -94.0364 -0.0207 -0.6607 31.8960 yr_rnd | -19.88875 -2.148 0.032 -8.4174 -0.1397 -0.0591 0.4232mobility | -1.30135 -2.983 0.003 -9.7692 -0.0091 -0.0686 7.5069 acs_k3 | 1.31870 0.585 0.559 1.8117 0.0093 0.0127 1.3738 acs_46 | 2.03246 2.546 0.011 7.8245 0.0143 0.0550 3.8498 full | 0.60972 1.281 0.201 9.0801 0.0043 0.0638 14.8924 emer | -0.70662 -1.167 0.244 -8.2569 -0.0050 -0.0580 11.6851 enroll | -0.01216 -0.724 0.469 -2.7548 -0.0001 -0.0194 226.4732---------------------------------------------------------------------------

Let us compare the regressoutput with the listcoefoutput. You will notice that the values listed in the Coef., t , and P>|t| values are the same in the twooutputs. The values listed in the Beta column of the regressoutput are the same as the values in the bStadXY column of listcoef. The bStdX column gives

the unit change in Y expected with a one standard deviation change in X. The bStdY column gives the standard deviation change in Y expected with a one

unit change in X. The SDofX column gives that standard deviation of each predictor variable in the model.

For example, the bStdX for ellis -21.3, meaning that a one standard deviation increase in ellwould lead to an expected 21.3 unit decrease in api00. The

bStdY value for ellof -0.0060 means that for a one unit, one percent, increase in english language learners, we would expect a 0.006 standard deviationdecrease in api00. Because the bStdX values are in standard units for the predictor variables, you can use these coefficients to compare the relative

strength of the predictors like you would compare Beta coefficients. The difference is BStdX coefficients are interpreted as changes in the units of the

outcome variable instead of in standardized units of the outcome variable. For example, the BStdX for mealsversus ellis -94 versus -21, or about 4 timesas large, the same ratio as the ratio of the Beta coefficients. We have created an annotated outputthat more thoroughly explains the output from listcoef.

So far, we have concerned ourselves with testing a single variable at a time, for example looking at the coefficient for elland determining if that is significant.

We can also test sets of variables, using the testcommand, to see if the set of variables are significant. First, let's start by testing a single variable, ell, usingthe testcommand.

test ell==0


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( 1) ell = 0.0

F( 1, 385) = 16.67 Prob > F = 0.0001

If you compare this output with the output from the last regression you can see that the result of the F-test, 16.67, is the same as the square of the result of

the t-test in the regression (-4.083^2 = 16.67). Note that you could get the same results if you typed the following since Stata defaults to comparing the

term(s) listed to 0.

test ell

( 1) ell = 0.0

F( 1, 385) = 16.67 Prob > F = 0.0001

Perhaps a more interesting test would be to see if the contribution of class size is significant. Since the information regarding class size is contained in twovariables, acs_k3and acs_46, we include both of these with the testcommand.

test acs_k3 acs_46

( 1) acs_k3 = 0.0( 2) acs_46 = 0.0

F( 2, 385) = 3.95 Prob > F = 0.0200

The significant F-test, 3.95, means that the collective contribution of these two variables is significant. One way to think of this, is that there is a significantdifference between a model with acs_k3and acs_46as compared to a model without them, i.e., there is a significant difference between the "full" model and

the "reduced" models.

Finally, as part of doing a multiple regression analysis you might be interested in seeing the correlations among the variables in the regression model. Youcan do this with the correlatecommand as shown below.

correlate api00 ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll

(obs=395)

| api00 ell meals yr_rnd mobility acs_k3 acs_46

-------------+--------------------------------------------------------------- api00 | 1.0000 ell | -0.7655 1.0000 meals | -0.9002 0.7711 1.0000 yr_rnd | -0.4831 0.5104 0.4247 1.0000 mobility | -0.2106 -0.0149 0.2207 0.0321 1.0000 acs_k3 | 0.1712 -0.0553 -0.1888 0.0222 0.0397 1.0000 acs_46 | 0.2340 -0.1743 -0.2137 -0.0419 0.1280 0.2708 1.0000 full | 0.5759 -0.4867 -0.5285 -0.4045 0.0235 0.1611 0.1212 emer | -0.5902 0.4824 0.5402 0.4401 0.0612 -0.1111 -0.1283 enroll | -0.3221 0.4149 0.2426 0.5920 0.1007 0.1084 0.0281

| full emer enroll-------------+--------------------------- full | 1.0000 emer | -0.9059 1.0000 enroll | -0.3384 0.3417 1.0000

If we look at the correlations with api00, we see mealsandellhave the two strongest correlations with api00. These correlations are negative, meaning thatas the value of one variable goes down, the value of the other variable tends to go up. Knowing that these variables are strongly associated with api00, we

might predict that they would be statistically significant predictor variables in the regression model.

We can also use the pwcorrcommand to do pairwise correlations. The most important difference between correlateand pwcorris the way in which missingdata is handled. With correlate, an observation or case is dropped if any variable has a missing value, in other words, correlateuses listwise , also called

casewise, deletion. pwcorruses pairwise deletion, meaning that the observation is dropped only if there is a missing value for the pair of variables being

correlated. Two options that you can use with pwcorr, but not with correlate, are the sigoption, which will give the significance levels for the correlationsand the obsoption, which will give the number of observations used in the correlation. Such an option is not necessary with corras Stata lists the number of

observations at the top of the output.

pwcorr api00 ell meals yr_rnd mobility acs_k3 acs_46 full emer enroll, obs sig

| api00 ell meals yr_rnd mobility acs_k3 acs_46-------------+---------------------------------------------------------------


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api00 | 1.0000|

| 400 | ell | -0.7676 1.0000

| 0.0000 | 400 400 | meals | -0.9007 0.7724 1.0000

| 0.0000 0.0000 | 400 400 400 |

yr_rnd | -0.4754 0.4979 0.4185 1.0000| 0.0000 0.0000 0.0000 | 400 400 400 400 | mobility | -0.2064 -0.0205 0.2166 0.0348 1.0000

| 0.0000 0.6837 0.0000 0.4883 | 399 399 399 399 399 | acs_k3 | 0.1710 -0.0557 -0.1880 0.0227 0.0401 1.0000

| 0.0006 0.2680 0.0002 0.6517 0.4245 | 398 398 398 398 398 398 | acs_46 | 0.2329 -0.1733 -0.2131 -0.0421 0.1277 0.2708 1.0000

| 0.0000 0.0005 0.0000 0.4032 0.0110 0.0000 | 397 397 397 397 396 395 397 | full | 0.5744 -0.4848 -0.5276 -0.3977 0.0252 0.1606 0.1177

| 0.0000 0.0000 0.0000 0.0000 0.6156 0.0013 0.0190

| 400 400 400 400 399 398 397 | emer | -0.5827 0.4722 0.5330 0.4347 0.0596 -0.1103 -0.1245

| 0.0000 0.0000 0.0000 0.0000 0.2348 0.0277 0.0131 | 400 400 400 400 399 398 397 | enroll | -0.3182 0.4030 0.2410 0.5918 0.1050 0.1089 0.0283

| 0.0000 0.0000 0.0000 0.0000 0.0360 0.0298 0.5741 | 400 400 400 400 399 398 397 |

| full emer enroll-------------+--------------------------- full | 1.0000

| | 400 | emer | -0.9057 1.0000

| 0.0000 | 400 400 | enroll | -0.3377 0.3431 1.0000

| 0.0000 0.0000 | 400 400 400 |

1.5 Transforming Variables

Earlier we focused on screening your data for potential errors. In the next chapter, we will focus on regression diagnostics to verify whether your data meetthe assumptions of linear regression. Here, we will focus on the issue of normality. Some researchers believe that linear regression requires that the

outcome (dependent) and predictor variables be normally distributed. We need to clarify this issue. In actuality, it is the residuals that need to be normallydistributed. In fact, the residuals need to be normal only for the t-tests to be valid. The estimation of the regression coefficients do not require normally

distributed residuals. As we are interested in having valid t-tests, we will investigate issues concerning normality.

A common cause of non-normally distributed residuals is non-normally distributed outcome and/or predictor variables. So, let us explore the distribution of

our variables and how we might transform them to a more normal shape. Let's start by making a histogram of the variable enroll, which we looked at earlierin the simple regression.

histogram enroll


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We can use the normaloption to superimpose a normal curve on this graph and the bin(20) option to use 20 bins. The distribution looks skewed to the right.

histogram enroll, normal bin(20)

You may also want to modify labels of the axes. For example, we use the xlabel()option for labeling the x-axis below, labeling it from 0 to 1600 incrementingby 100.

histogram enroll, normal bin(20) xlabel(0(100)1600)

Histograms are sensitive to the number of bins or columns that are used in the display. An alternative to histograms is the kernel density plot, which

approximates the probability density of the variable. Kernel density plots have the advantage of being smooth and of being independent of the choice of


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origin, unlike histograms. Stata implements kernel density plots with the kdensitycommand.

kdensity enroll, normal

Not surprisingly, the kdensityplot also indicates that the variable enrolldoes not look normal. Now let's make a boxplot for enroll, using graph box

command.

graph box enroll

Note the dots at the top of the boxplot which indicate possible outliers, that is, these data points are more than 1.5*(interquartile range) above the 75th

percentile. This boxplot also confirms that enrollis skewed to the right.

There are three other types of graphs that are often used to examine the distribution of variables; symmetry plots, normal quantile plots and normal

probability plots.

A symmetry plot graphs the distance above the median for the i-th value against the distance below the median for the i-th value. A variable that is symmetricwould have points that lie on the diagonal line. As we would expect, this distribution is not symmetric.

symplot enroll


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A normal quantile plot graphs the quantiles of a variable against the quantiles of a normal (Gaussian) distribution. qnormis sensitive to non-normality nearthe tails, and indeed we see considerable deviations from normal, the diagonal line, in the tails. This plot is typical of variables that are strongly skewed to the

right.

qnorm api00

Finally, the normal probability plot is also useful for examining the distribution of variables. pnorm is sensitive to deviations from normality nearer to the

center of the distribution. Again, we see indications of non-normality in enroll.

pnorm enroll

Having concluded that enrollis not normally distributed, how should we address this problem? First, we may try entering the variable as-is into the


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regression, but if we see problems, which we likely would, then we may try to transform enrollto make it more normally distributed. Potential transformations

include taking the log, the square root or raising the variable to a power. Selecting the appropriate transformation is somewhat of an art. Stata includes the

ladderand gladdercommands to help in the process. Ladderreports numeric results and gladderproduces a graphic display. Let's start with ladderandlook for the transformation with the smallest chi-square.

ladder enroll

ladder enroll

Transformation formula chi2(2) P(chi2)------------------------------------------------------------------cube enroll^3 . 0.000square enroll^2 . 0.000raw enroll . 0.000square-root sqrt(enroll) 20.56 0.000log log(enroll) 0.71 0.701reciprocal root 1/sqrt(enroll) 23.33 0.000reciprocal 1/enroll 73.47 0.000reciprocal square 1/(enroll^2) . 0.000reciprocal cube 1/(enroll^3) . 0.000

The log transform has the smallest chi-square. Let's verify these results graphically using gladder.

gladder enroll

This also indicates that the log transformation would help to make enrollmore normally distributed. Let's use the generatecommand with the logfunction to

create the variable lenrollwhich will be the log of enroll. Note that login Stata will give you the natural log, not log base 10. To get log base 10, typelog10(var).

generate lenroll = log(enroll)

Now let's graph our new variable and see if we have normalized it.

hist lenroll, normal


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We can see that lenrolllooks quite normal. We would then use the symplot, qnormand pnormcommands to help us assess whether lenrollseemsnormal, as well as seeing how lenrollimpacts the residuals, which is really the important consideration.

1.6 Summary

In this lecture we have discussed the basics of how to perform simple and multiple regressions, the basics of interpreting output, as well as some relatedcommands. We examined some tools and techniques for screening for bad data and the consequences such data can have on your results. Finally, we

touched on the assumptions of linear regression and illustrated how you can check the normality of your variables and how you can transform your variables

to achieve normality. The next chapter will pick up where this chapter has left off, going into a more thorough discussion of the assumptions of linear

regression and how you can use Stata to assess these assumptions for your data. In particular, the next lecture will address the following issues.

Checking for points that exert undue influence on the coefficients

Checking for constant error variance (homoscedasticity)

Checking for linear relationships

Checking model specification

Checking for multicollinearity

Checking normality of residuals

See the Stata Topics: Regressionpage for more information and resources on simple and multiple regression in Stata.

1.7 Self Assessment

Make five graphs of api99: histogram, kdensity plot, boxplot, symmetry plot and normal quantile plot.1.What is the correlation between api99and meals?2.

Regress api99on meals. What does the output tell you?3.

Create and list the fitted (predicted) values.4.

Graph mealsand api99with and without the regression line.5.

Look at the correlations among the variables api99mealsellavg_edusing the corrand pwcorrcommands. Explain how these commands are

different. Make a scatterplot matrix for these variables and relate the correlation results to the scatterplot matrix.

6.

Perform a regression predicting api99from meals andell. Interpret the output.7.

Click herefor our answers to these self assessment questions.

1.8 For More Information

Stata Manuals

[R] regress

[R] predict

[R] test

Related Web Pages

Stata FAQ- How can I do a scatterplot with regression line in Stata?

Annotated Stata Output- Regression

Stata Add On Programs


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Regression With Stata: Lesson 1 - Simple and Multiple Regression

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